How can I “fix” my genuine damaged brown leather couch?

You could try brown shoe polish, then buff well, just as you would use on shoes or worn purses.

Leather dye.

I've bee fixing scratched leather for years with shoe polish. This looks like you will have to mix three colors to have the scratches not show. I think just brown would be too dark.

I've never used the dye so don't know about that.

Here are some before and after photos. This has very kittle brown in the mix. I used Tan and Cordovan with a touch of brown.

Don't be cheap or silly! Leather is producted using leather dye which can be purchased online from a variety of sources in a range of colors. Carpet cleaners, leather cleaners, upholstery cleaners, etc. use leather dye to redye furniture and to repair leather furniture. As with any quality workmanship it takes patience. Most leather furniture starts to appear worn and scratched due to leather drying out, moisturize your couch just as you would your own skin which of course is also leather. A little dye on the scratch and moisturizer can bring a old leather couch back to life.

I looked around the innerwebz & found a few kits for your leather restoration project, Ize31332293 ,but they look like a better fit for a professional. Lots of stuff you don't need & don't need to store either. I narrowed it down to one for you

This company has premixed colors, one of which is probably close enough. Perhaps F36 Chestnut would work. See what you think.

http://www.leatherworldtech.com/Leather-Furniture-Dye-Color-Chart-s/1875.htm

There are some how to videos there too. Best to you! :-)